1. Field of the Invention
The present invention relates to systems and methods for detection and measurement of circular birefringences in materials, such as optically active (chiral) liquids and materials that exhibit the Faraday effect.
2. Brief Description of the Related Art
An isotropic optically active medium, such as a chiral liquid, is characterized by a difference in the refractive indices for left- and right-circularly polarized light. This refractive index difference is known as circular birefringence and gives rise to the rotation of the plane of polarization of a linearly polarized light beam traversing the medium. Optical rotation only arises in an isotropic medium, such as a liquid, if the medium is chiral. The measurement of optical rotation in an isotropic medium (in the absence of a static magnetic field) is therefore a measure of the presence of chiral molecules and is the basis of standard laboratory instruments such as polarimeters. The optical rotation measured in polarimeters is directly proportional to the distance the light traverses through the sample. See Barron, L. D. Molecular light scattering and optical activity (Cambridge University Press, Cambridge, 2004).
There is a need to measure optical activities—such as those that are due to chiral molecules—without the need for long path-lengths, e.g. in small volumes. By measuring a property that is a function of the relative difference in the direction of propagation of the two circularly polarized light components, rather than the rotation of the plane of polarization, the present invention shows that optical activities can be determined in an alternate way that is not a function of the distance the light traverses through the sample. An apparatus based on the principle detailed in the present invention may therefore be used to determine the optical activity, enantiomeric excess, optical purity, chemical composition, etc. of a small volume of liquid, gas, or solid, and can therefore form the basis for a detector of, say chiral analytes in suitable capillaries, liquid drops etc.
The invention is based on a difference in the angles of refraction between the two circularly polarized light components that refract at an interface in the presence of circular birefringence and/or a difference in the angles of reflection between the two circularly polarized light components that reflect at an interface formed by the medium that exhibits circular birefringence and a suitable reflector. Similar phenomena may also be observed in diffraction. First, these principles are applied to the measurement of natural optical activity (chiral liquids), then it is shown how the same principles also apply to optical activity as is induced by a magnetic field (Faraday effect).
In general, light refracts and its speed and direction can change as it traverses the boundary between two (at least partly) transparent media. Should a material be characterized by polarization-dependent refractive indices, then the different polarization components of a wave refract differently. This phenomenon, known as double refraction or birefringence, is found in many anisotropic media, such as crystals, where it may cause a ray of light to separate into two. See Born, M. & Wolf, E. Principles of Optics (Cambridge University Press, Cambridge, 1999) and Ditchburn, R. W. Light (Dover, N.Y., 1991). Isotropic media, such as a liquid or a gas, can only give rise to double refraction if they are distinct from their mirror image (are chiral) and thus optically active. Because of a difference in refractive indices for left- and right-circularly polarized light, linearly polarized or unpolarized light entering an isotropic optically active medium at an angle, will split into two waves at the boundary, one left- and the other right-circularly polarized, with a small difference in the respective angles of refraction. See Fresnel, A. J. in Euvres complètes d' Augustin Fresnel (eds. Sénarmont, H. de, Verdet, É. & Fresnel, L.) (Paris, 1866); Fresnel, A. Ann. Chim. Phys. 28, 147 (1825); and Lowry, T. M. Optical rotatory power (Dover, N.Y., 1964). The doubling of a linear or unpolarized light beam into its circular polarization components after traversing the interface(s) formed by an optically active medium have been reported to have been observed, both via multiple-refractions in a chiral liquid (E. v. Fleischl, “Die doppelte Brechnung des Lichtes in Flüssigkeiten,”Sitz. Ber. Kais. Akad. Wiss. (Math-Nat) 90, (1884), 478), as well as via multiple-reflections in a Faraday medium (D. B. Brace, “On the resolution of light into its circular components in the Faraday effect”, Phil. Mag. 6, (1885), 464-475). Essentially a single beam of light is observed to double. This, however, is a relatively impractical and insensitive scheme to measure circular birefringences. Much more sensitive are the schemes that form the basis of the present invention.
The apparatus and methods of this invention show that instead of taking a picture of a single light beam forming a double image, it is more practical and sensitive to detect the position of a light beam on a position sensitive detector as the light is modulated between left- and right-circular polarized. The apparatus and methods of this invention can be observed at a single interface, do not require large sample volumes, and are readily implemented. (See Ghosh, A., and Fischer, P., “Chiral molecules split light: reflection and refraction in a chiral liquid”, Phys. Rev. Letters 97 (2007) 173002.)
It is shown that the difference in propagation directions (angular divergence) measured using the present invention can be used to sensitively determine the optical purity (enantiomeric excess) of a chiral liquid. The angular divergence between the two refracted circularly polarized waves contains information identical to that obtained from conventional optical rotation measurements. However, unlike optical rotation, which depends on the path length through the sample, chiral double refraction, as this phenomenon may be termed, arises within a few wavelengths from the boundary.
Isotropic media, such as liquids, composed of randomly oriented molecules are, in the absence of an external influence generally described by a single scalar refractive index. It follows that there is only one refracted beam and only one angle of refraction. This is given by Snell's law (See Hecht, E., “Optics”, Addison Wesley, 2004):n1 sin θ1=n2 sin θ2,  (1)where the angle the beam makes with the normal of the interface is θ1 in the medium with refractive index n1 and similarly θ2 is the angle the beam makes with the normal of the interface in the medium with n2. It therefore follows that light incident on an interface formed by two isotropic media 1 and 2 with angle of incidence θ1 in medium 1 will change its direction, i.e. refract in medium 2, where its angle of refraction is given by
                              θ          2                =                              arcsin            ⁡                          (                                                                    n                    1                                                        n                    2                                                  ⁢                sin                ⁢                                                                  ⁢                                  θ                  1                                            )                                .                                    (        2        )            
Optically active media have the power to rotate the polarization of a linearly polarized light beam. Of particular interest are media that have this property even when they are isotropic, i.e. in the absence of any strain, (quasi) static fields, or other perturbations that could cause the dielectric function to become direction dependent. Such optically active media that do not have a direction dependent dielectric function (refractive indices) and therefore do not have just one scalar refractive index, can be characterized by two refractive indices, one for left- (−) and one for right- (+) circularly polarized radiation. Note, that linearly polarized light may be regarded as a coherent superposition of left- and right-circularly polarized waves of equal amplitude, and Fresnel's theory of optical rotation shows that a difference in the respective refractive indices n(−) and n(+) causes the waves to acquire different phases and the polarization vector to rotate. The optical rotation α in radians developed by light at the wavelength λ traversing a distance d in an optically active medium, such as a chiral liquid, accordingly is:
                    α        =                                            π              ⁢                                                          ⁢              d                        λ                    ⁢                      (                                          n                                  (                  -                  )                                            -                              n                                  (                  +                  )                                                      )                                              (        3        )            and is a function of the circular birefringence n(−)−n(+). See Barron, L. D. Molecular light scattering and optical activity (Cambridge University Press, Cambridge, 2004) and Hecht, E., “Optics”, Addison Wesley, 2004. A significance of optical rotation lies in the fact that it is a means to distinguish the two mirror image forms (enantiomers) of a chiral molecule, since n(−)−n(+) and hence α is of opposite sign for the two enantiomers. Most biologically important molecules, such as sugars, are chiral and optical rotation is a well established analytical technique used to determine their absolute stereochemical configuration (and concentration) in solution. See Povalarapu, P. L. Chirality 14, 768-781 (2002).
Since isotropic optically active media possess two refractive indices, their double refraction can not only manifest itself through optical rotation, but also through a difference in the corresponding angles of refraction (and similarly diffraction in a suitable transmission grating). Each circularly polarized wave, or component of a wave, will bend differently at a boundary, and these refraction events are therefore independently described by expressions analogous to Eq. (1). For instance, refraction at an interface formed by an achiral medium, such as air, with refractive index n, and an optically active medium, such as a chiral liquid, with refractive indices n(+) and n(−) for right- and left-circularly polarized light, respectively will be described byn sin θ=n(+)sin θ(+),  (4)andn sin θ=n(−)sin θ(−),  (5)where θ is the angle of incidence in the achiral medium, and where θ(+) and θ(−) are the angles of refraction for the right- and left-circularly polarized light components respectively. An unpolarized or a linearly polarized light beam incident on an interface described by Eqs. (4) and (5) will thus split in the chiral medium into its circular components which will travel with different angles of refraction. This is depicted schematically in FIG. 1a. An unpolarized or linearly polarized light beam 10 is incident from a medium 11 with refractive index n onto the boundary (interface) 16 formed with medium 12 that is characterized by refractive indices n(+) and n(−). The normal of the interface is 13. The two refracted circularly polarized components are 14 and 15. Alternately, as shown in FIG. 2a, if right-circularly polarized light 20 is incident on this interface, formed by media 11 and 12, then it will refract and travel in the direction 21 given by angle of refraction θ(+). As seen in FIG. 2b, left-circularly polarized light 22 incident on this interface it will correspondingly take a different path 23 given by θ(−).
Such splitting and/or differences in angles of refraction are also present if the light travels from the chiral medium to the achiral medium, i.e. is incident from 12 onto 11 (an chiral/achiral interface). Similarly, such splitting and/or differences in angles of refraction can be observed if the light travels from one chiral medium to another chiral medium which is characterized by a different circular birefringence. As shown in FIG. 3a, an unpolarized or linearly polarized light beam 10 that is incident normal onto a prismatic sample 30 that exhibits circular birefringence will deviate in its direction of propagation and will separate into its two circularly polarized components 14 and 15 that correspondingly exhibit an angular divergence. Should two prismatic cuvettes 30 be arranged as shown in FIG. 3b and such that 10 traverses their interface at an angle, then the two circularly polarized components 14 and 15 diverge symmetrically about 10 if the two media 31 and 32 in 30 have the same scalar refractive index (n(−)+n(+))/2, but differ only in their circular birefringence (pseudoscalar refractive index). This may for instance be achieved if the liquids in 31 and 32 only differ in their respective amounts of enantiomeric excess.
In the preferred embodiments of the present invention that are based on refraction and/or reflection the light must not be incident normal to the interface, i.e. along 13, in FIGS. 1 and 2.
The invention details how the determination of the difference between these angles of refraction and/or reflection, i.e. θ(−)−θ(+), or any related such measurement, can be used as an alternative to optical rotation measurements and as a diagnostic for optical activity and therefore chirality.
A difference in the angles of reflection can occur in an optically active medium, as schematically shown in FIG. 1b, if the reflection preserves (at least some) of the circularity of the light upon reflection. Linearly polarized or unpolarized light 10 is incident from an optically active medium 110 characterized by unequal refractive indices for left and right-circularly polarized light n(−) and n(+). 10 is incident with angle of incidence θ upon a reflecting surface 17. The circular polarization components change sign upon reflection, i.e. a left-circularly polarized component will become (at least partially) right-circularly polarized upon reflection and therefore experiences a different refractive index if the medium is circular birefringent. The angle of reflection is θrefl.(±) for left (−) and right (+) circularly polarized light is given by
                              θ                      refl            .                                (            ±            )                          =                              arcsin            ⁡                          (                                                                    n                                          (                      ∓                      )                                                                            n                                          (                      ±                      )                                                                      ⁢                sin                ⁢                                                                  ⁢                θ                            )                                .                                    (        6        )            
As seen in FIG. 1c, left-circularly polarized 101 and right-circularly polarized light 103 incident upon a reflecting surface 17 with the same angle of incidence θ will reflect with different angles of reflections (given by Eq. 6) and therefore propagate with different directions 102 and 104 respectively.
The invention is also applicable in situations where diffraction in an optically active medium is important. The diffraction phenomena are related to the refraction and reflection phenomena described above. For a diffraction grating of groove spacing D, the angular position, θ of the diffracted spot is given by (See Hecht, E., “Optics”, Addison Wesley, 2004):
                              D          ⁢                                          ⁢          sin          ⁢                                          ⁢          θ                =                  m          ⁢                                                    λ                0                            n                        .                                              (        7        )            Here m is the order of diffraction, λ0 is the vacuum wavelength of light and n is the refractive index of the light in the medium where the diffraction grating is placed. Since the refractive indices of right and left circularly polarized light are different in an optically active medium, the angular positions of the diffracted spots are also different. The angular deviation between the right and left circularly polarized light after diffraction is given by (Ambarish Ghosh, Furqan Fazal, and Peer Fischer, “Circular differential double diffraction in chiral media”, Optics Letters, (2007), Doc. ID 81432):
                                          θ                          (              +              )                                -                      θ                          (              -              )                                      =                              arcsin            ⁡                          (                                                m                  ⁢                                                                          ⁢                                      λ                    0                                                                    Dn                                      (                    +                    )                                                              )                                -                      arcsin            ⁡                          (                                                m                  ⁢                                                                          ⁢                                      λ                    0                                                                    Dn                                      (                    -                    )                                                              )                                                          (        8        )            Consequently, a measurement of the angular deviation between the two light beams provides a direct measure of the optical activity of the liquid. This or a similar analysis applies for diffraction in transmission or reflection.
The application of a magnetic field in the direction of propagation of a light beam renders all media optically active, i.e. causes circular birefringence. However, these media are also anisotropic. The anisotropy means that the light propagation is in general described by effects in addition to those that are due to circular birefringence. Nevertheless, if these additional effects may be neglected, or if they are small, or if the magnetic field has a component along the direction of propagation of the light beam, then phenomena essentially similar to those due to chirality and described above can be observed (“Observation of the Faraday effect via beam deflection in a longitudinal magnetic field”, Ambarish Ghosh, and Peer Fischer, eprint arXiv:physics/0702063, 2007). The magnetic field induced effects differ from the effects in chiral liquids in that they are unrelated to molecular chirality and in that they depend on the relative direction between propagation direction and magnetic field direction. However, for a field along the propagation direction, the circular birefringence induced is given by
                    α        =                                                            π                ⁢                                                                  ⁢                d                            λ                        ⁢                          (                                                n                                      (                    -                    )                                                  -                                  n                                      (                    +                    )                                                              )                                =          VBd                                    (        9        )            where V is the Verdet constant and B the magnetic field strength. The difference in the angles of refraction and reflection and diffraction for circularly polarized light components due to the Faraday effect are essentially the same as those (see above) obtained for chiral samples. See also (“Observation of the Faraday effect via beam deflection in a longitudinal magnetic field”, Ambarish Ghosh, and Peer Fischer, eprint arXiv:physics/0702063, 2007).